Chemotactic effects in reaction-diffusion equations for inflammation.

Predator-prey systems are used to model time-dependent virus and lymphocyte population during a liver infection and to discuss the influence of chemotactic behavior on the chronification tendency of such infections. Therefore, a model family of reaction-diffusion equations is presented, and the long-term behavior of the solutions is estimated by a critical value containing the reaction strength, the diffusion rate, and the extension of the liver domain. Fourier techniques are applied to evaluate the influence of chemotactic behavior of the immune response to the long-term behavior of locally linearized models. It turns out that the chemotaxis is a subordinated influence with respect to the chronification of liver infections.

Modeling the Chronification Tendency of Liver Infections as Evolutionary Advantage.

Here, we discuss how the tendency of a liver infection to chronify can be seen as an evolutionary advantage for infected individuals. For this purpose, we present a set of reaction-diffusion equations as a mathematical model of viral liver infections, which allows chronic and acute courses of the liver infection. We introduce a cumulative wealth function, and finally, we show that an immune response favoring the chronification is evolutionary advantageous at the same time.

LTP or LTD? Modeling the Influence of Stress on Synaptic Plasticity.

In cognitive memory, long-term potentiation (LTP) has been shown to occur when presynaptic and postsynaptic activities are highly correlated and glucocorticoid concentrations are in an optimal (i.e., low normal) range. In all other conditions, LTP is attenuated or even long-term depression (LTD) occurs. In this paper, we focus on NMDA receptor (NMDA-R)-dependent LTP and LTD, two processes involving various molecular mechanisms. To understand which of these mechanisms are indispensable for explaining the experimental evidence reported in the literature, we here propose a parsimonious model of NMDA-R-dependent synaptic plasticity. Central to this model are two processes. First, AMPA receptor-subunit trafficking; and second, glucocorticoid-dependent modifications of the brain-derived neurotrophic factor (BDNF)-receptor system. In 2008, we have published a core model, which contained the first process, while in the current paper we present an extended model, which also includes the second process. Using the extended model, we could show that stress attenuates LTP, while it enhances LTD. These simulation results are in agreement with experimental findings from other labs. In 2013, surprising experimental evidence showed that the GluA1 C-tail is unnecessary for LTP. When using our core model in its original form, our simulations already predicted that there would be no requirement for the GluA1 C-tail for LTP, allowing to eliminate a redundant mechanism from our model. In summary, we present a mathematical model that displays reduced complexity and is useful for explaining when and how LTP or LTD occurs at synapses during cognitive memory formation.

Genetics of metabolic resistance.

Herbicide resistance has become a major issue for many weeds. Metabolic resistance refers to the biochemical processes within organisms that degrade herbicides to less toxic compounds, resulting in a shift of the dose response curve. This type of resistance involves polygenic inheritance. A model is presented linking the biochemical pathway of amino acid synthesis and the detoxifying pathway of an inhibitor of the key enzyme ALS. From this model, resistance factors for each biotype are derived, which are then applied to a polygenic population genetic model for an annual weed plant. Polygenic inheritance is described by a new approach based on tensor products of heredity matrices. Important results from the model are that low dose regimes favour fast emergence of resistant biotypes and that the emergence of resistant biotypes occurs as abrupt outbreaks. The model is used to evaluate strategies for the management of metabolic resistance.

Modelling and analysis of a gene-regulatory feed-forward loop with basal expression of the second regulator.

Efficient adaptation strategies to changing environmental conditions are essential for bacteria to survive and grow. Fundamental restructuring of their metabolism is usually mediated by corresponding gene regulation. Here, often several different environmental stimuli have to be integrated into a reasonable, energy-efficient response. Fast fluctuations and overshooting have to be filtered out. The gene regulatory network for the anaerobic adaptation of the pathogenic bacterium Pseudomonas aeruginosa is organized as a feed-forward loop (FFL), which is a three-gene network motif composed of two transcription factors (Anr for oxygen, NarxL for nitrate) and one target (Nar for nitrate reductase). The upstream transcription factor (Anr) induces the downstream transcription factor (NarXL). Both regulators act together positively by inducing the target (Nar) via a direct and indirect regulation path (coherent type-1 FFL). Since full promoter activity is only achieved when both transcription factors are present the target operon is expressed with a delay. Thus, in response to environmental stimuli (oxygen, nitrate), signals are mediated and processed in a way that short pulses are filtered out. In this study we analyze a special kind of FFL called FFLk by means of a family of ordinary differential equation models. The secondary FFL regulator (NarXL) is expressed constitutively but further induced in the presence of the upstream stimuli. This FFL modification has substantial influence on the response time and cost-benefit ratio mediated by environmental fluctuations. In order to find conditions where this regulatory network motif might be beneficial, we analyzed various models and environments. We describe the observed evolutional advantage of FFLk and its role in environmental adaptation and pathogenicity.

Energy allocation between brain and body during ontogenetic development.

OBJECTIVE: We here studied how energy is allocated between brain and body both during the ontogenetic development from a child to an adult and during weight loss. METHODS: We investigated 180 normal weight female and male children and adolescents (aged 6.1-19.9 years) as well as 35 overweight adolescents undergoing weight reduction intervention. 52 normal weight and 42 obese adult women were used for comparison. We assessed brain mass by magnetic-resonance-imaging and body metabolism by indirect calorimetry. To study how energy is allocated between brain and body, we measured plasma insulin, since insulin fulfils the functions of a glucose allocating hormone, i.e., peripheral glucose uptake depends on insulin, central uptake does not. We used reference data obtained in the field of comparative biology. In a brain-body-plot, we calculated the distance between each subject and a reference mammal of comparable size and named the distance "encephalic measure." With higher encephalic measures, more energy is allocated to the brain. RESULTS: We found that ontogenetic development from a child to an adult was indicated by decreasing encephalic measures in females (r = -0.729, P < 0.001) and increasing plasma insulin concentrations (F = 6.6, P = 0.002 in females and F = 8.6, P < 0.001 in males). Weight loss of about 5 kg in females and about 9 kg in males resulted in reduced insulin concentrations and increased encephalic measures. CONCLUSION: Our results indicate that the share of energy allocated to the brain increased with weight loss, but decreased during the ontogenetic development from childhood to adolescence. These developmental changes in brain-to-body energy allocation appear to be driven by increasing plasma insulin concentrations.

The corpulent phenotype-how the brain maximizes survival in stressful environments.

The reactivity of the stress system may change during the life course. In many-but not all-humans the stress reactivity decreases, once the individual is chronically exposed to a stressful and unsafe environment (e.g., poverty, work with high demands, unhappy martial relationship). Such an adaptation is referred to as habituation. Stress habituation allows alleviating the burden of chronic stress, particularly cardiovascular morbidity and mortality. Interestingly, two recent experiments demonstrated low stress reactivity during a mental or psychosocial challenge in subjects with a high body mass. In this focused review we attempt to integrate these experimental findings in a larger context. Are these data compatible with data sets showing a prolonged life expectancy in corpulent people? From the perspective of neuroenergetics, we here raise the question whether "obesity" is unhealthy at all. Is the corpulent phenotype possibly the result of "adaptive phenotypic plasticity" allowing optimized survival in stressful environments?

Multi-gene-loci inheritance in resistance modeling.

The ability of an organism to degrade harmful substances to less toxic compounds is referred to as metabolic resistance. The biochemical processes result in a shift of dose-response curves associated with the toxic substances. Hence, the development of metabolic resistance may cause great problems of managing pests and diseases by pesticides. We develop a polygenic fitness model capable of simulating the emergence of metabolic resistance. Within the model, polygenic inheritance is described by a new approach based on tensor products of heredity matrices. This is included as genetic submodel into the time-continuous population model for all possible biotypes. Evolution is acting on the parameters of dose-response curves, i.e., on the mortality rates and thus on the ED(50)-value. The resulting system of differential equations is analyzed with respect to polymorphic equilibria. Under a longterm application of only one mode of action the model produces a gradual shift of the mean dose-response curve of the population which is frequently observed in the field. Different scenarios of the development of metabolic resistance are demonstrated in numerical experiments.

The brain's supply and demand in obesity.

During psychosocial stress, the brain demands extra energy from the body to satisfy its increased needs. For that purpose it uses a mechanism referred to as "cerebral insulin suppression" (CIS). Specifically, activation of the stress system suppresses insulin secretion from pancreatic beta-cells, and in this way energy-particularly glucose-is allocated to the brain rather than the periphery. It is unknown, however, how the brain of obese humans organizes its supply and demand during psychosocial stress. To answer this question, we examined 20 obese and 20 normal weight men in two sessions (Trier Social Stress Test and non-stress control condition followed by either a rich buffet or a meager salad). Blood samples were continuously taken and subjects rated their vigilance and mood by standard questionnaires. First, we found a low reactive stress system in obesity. While obese subjects showed a marked hormonal response to the psychosocial challenge, the cortisol response to the subsequent meal was absent. Whereas the brains of normal weight subjects demanded for extra energy from the body by using CIS, CIS was not detectable in obese subjects. Our findings suggest that the absence of CIS in obese subjects is due to the absence of their meal-related cortisol peak. Second, normal weight men were high reactive during psychosocial stress in changing their vigilance, thereby increasing their cerebral energy need, whereas obese men were low reactive in this respect. Third, normal weight subjects preferred carbohydrates after stress to supply their brain, while obese men preferred fat and protein instead. We conclude that the brain of obese people organizes its need, supply, and demand in a low reactive manner.

Effect of constitution on mass of individual organs and their association with metabolic rate in humans--a detailed view on allometric scaling.

Resting energy expenditure (REE)-power relationships result from multiple underlying factors including weight and height. In addition, detailed body composition, including fat free mass (FFM) and its components, skeletal muscle mass and internal organs with high metabolic rates (i.e. brain, heart, liver, kidneys), are major determinants of REE. Since the mass of individual organs scales to height as well as to weight (and, thus, to constitution), the variance in these associations may also add to the variance in REE. Here we address body composition (measured by magnetic resonance imaging) and REE (assessed by indirect calorimetry) in a group of 330 healthy volunteers differing with respect to age (17-78 years), sex (61% female) and BMI (15.9-47.8 kg/m(2)). Using three dimensional data interpolation we found that the inter-individual variance related to scaling of organ mass to height and weight and, thus, the constitution-related variances in either FFM (model 1) or kidneys, muscle, brain and liver (model 2) explained up to 43% of the inter-individual variance in REE. These data are the first evidence that constitution adds to the complexity of REE. Since organs scale differently as weight as well as height the "fit" of organ masses within constitution should be considered as a further trait.

The selfish brain: stress and eating behavior.

The brain occupies a special hierarchical position in human energy metabolism. If cerebral homeostasis is threatened, the brain behaves in a "selfish" manner by competing for energy resources with the body. Here we present a logistic approach, which is based on the principles of supply and demand known from economics. In this "cerebral supply chain" model, the brain constitutes the final consumer. In order to illustrate the operating mode of the cerebral supply chain, we take experimental data which allow assessing the supply, demand and need of the brain under conditions of psychosocial stress. The experimental results show that the brain under conditions of psychosocial stress actively demands energy from the body, in order to cover its increased energy needs. The data demonstrate that the stressed brain uses a mechanism referred to as "cerebral insulin suppression" to limit glucose fluxes into peripheral tissue (muscle, fat) and to enhance cerebral glucose supply. Furthermore psychosocial stress elicits a marked increase in eating behavior in the post-stress phase. Subjects ingested more carbohydrates without any preference for sweet ingredients. These experimentally observed changes of cerebral demand, supply and need are integrated into a logistic framework describing the supply chain of the selfish brain.

Why doesn't the brain lose weight, when obese people diet?

OBJECTIVE: As has been shown recently, obesity is associated with brain volume deficits. We here used an interventional study design to investigate whether the brain shrinks after caloric restriction in obesity. To elucidate mechanisms of neuroprotection we assessed brain-pull competence, i.e. the brain's ability to properly demand energy from the body. METHODS: In 52 normal-weight and 42 obese women (before and after ≈10% weight loss) organ masses of brain, liver and kidneys (magnetic resonance imaging), fat (air displacement plethysmography) and muscle mass (dual-energy X-ray absorptiometry) were assessed. Body metabolism was measured by indirect calorimetry. To investigate how energy is allocated between brain and body, we used reference data obtained in the field of comparative biology. We calculated the distance between each woman and a reference mammal of comparable size in a brain-body plot and named the distance 'encephalic measure'. To elucidate how the brain protects its mass, we measured fasting insulin, since 'cerebral insulin suppression' has been shown to function as a brain-pull mechanism. RESULTS: Brain mass was equal in normal-weight and obese women (1,441.8 ± 14.6 vs. 1,479.2 ± 12.8 g; n.s.) and was unaffected by weight loss (1,483.8 ± 12.7 g; n.s.). In contrast, masses of muscle, fat, liver and kidneys decreased by 3-18% after weight loss (all p < 0.05). The encephalic measure was lower in obese than normal-weight women (5.8 ± 0.1 vs. 7.4 ± 0.1; p < 0.001). Weight loss increased the encephalic measure to 6.3 ± 0.1 (p < 0.001). Insulin concentrations were inversely related to the encephalic measure (r = -0.382; p < 0.001). CONCLUSION: Brain mass is normal in obese women and is protected during caloric restriction. Our data suggest that neuroprotection during caloric restriction is mediated by a competent brain-pull exerting cerebral insulin suppression.

Systemic investigation of a brain-centered model of the human energy metabolism.

The regulation of the human energy metabolism is crucial to ensure the functionality of the entire organism. Deregulations may lead to severe pathologies such as diabetes mellitus and obesity. The decisive role of the brain as active controller and heavy consumer in the complex whole-body energy metabolism is the object of recent research. Latest studies suggest the priority of the brain energy supply in the competition between brain and body periphery for the available energy resources. In this paper, a systemic investigation of the human energy metabolism is presented which consists of a compartment model including periphery, blood, and brain as well as signaling paths via insulin, appetite, and ingestion. The presented dynamical system particularly contains the competition for energy between brain and body periphery. Characteristically, the hormone insulin is regarded as central feedback signal of the brain. The model realistically reproduces the qualitative behavior of the energy metabolism. Short-time observations demonstrate the physiological periodic food intake generating the typical oscillating blood glucose variations. Integration over the daily cycle yields a long-term model which shows a stable behavior in accordance with the homeostatic regulation of the energy metabolism on a long-time scale. Two types of abstract constitutive equations describing the interaction between compartments and signals are taken into consideration. These are nonlinear and linear representatives from the class of feasible relations. The robustness of the model against the choice of the representative relation is linked to evolutionary stability of existing organisms.

Stress and eating behavior.

How stress, the stress response, and the adaptation of the stress response influence our eating behavior is a central question in brain research and medicine. In this report, we highlight recent advances showing the close links between eating behavior, the stress system, and neurometabolism.

Compact energy metabolism model: brain controlled energy supply.

The regulation of the energy metabolism is crucial to ensure the functionality of the entire organism. Deregulations may lead to severe pathologies such as obesity and type 2 diabetes mellitus. The decisive role of the brain as the active controller and heavy consumer in the complex whole body energy metabolism is the matter of recent research. Latest studies suggest that the brain's energy supply has the highest priority while all organs in the organism compete for the available energy resources. In our novel mathematical model, we address these new findings. We integrate energy fluxes and their control signals such as glucose fluxes, insulin signals as well as the ingestion momentum in our new dynamical system. As a novel characteristic, the hormone insulin is regarded as central feedback signal of the brain. Hereby, our model particularly contains the competition for energy between brain and body periphery. The analytical investigation of the presented dynamical system shows a stable long-term behavior of the entire energy metabolism while short time observations demonstrate the typical oscillating blood glucose variations as a consequence of food intake. Our simulation results demonstrate a realistic behavior even in situations like exercise or exhaustion, and key elements like the brain's preeminence are reflected. The presented dynamical system is a step towards a systemic understanding of the human energy metabolism and thus may shed light to defects causing diseases based on deregulations in the energy metabolism.

Unbuffered and buffered supply chains in human metabolism.

The investigation of very complex dynamical systems like the human metabolism requires the comprehension of important subsystems. The present paper deals with energy supply chains as subsystems of the metabolism on the molecular, cellular, and individual levels. We form a mathematical model of ordinary differential equations and we show fundamental properties by Fourier techniques. The results are supported by a transition from a system of ordinary differential equations to a partial differential equation, namely, a transport equation. In particular, the behavior of supply chains with dominant pull components is discussed. A special focus lies on the role of buffer compartments.

Build-ups in the supply chain of the brain: on the neuroenergetic cause of obesity and type 2 diabetes mellitus.

Obesity and type 2 diabetes have become the major health problems in many industrialized countries. A few theoretical frameworks have been set up to derive the possible determinative cause of obesity. One concept views that food availability determines food intake, i.e. that obesity is the result of an external energy "push" into the body. Another one views that the energy milieu within the human organism determines food intake, i.e. that obesity is due to an excessive "pull" from inside the organism. Here we present the unconventional concept that a healthy organism is maintained by a "competent brain-pull" which serves systemic homeostasis, and that the underlying cause of obesity is "incompetent brain-pull", i.e. that the brain is unable to properly demand glucose from the body. We describe the energy fluxes from the environment, through the body, towards the brain with a mathematical "supply chain" model and test whether its predictions fit medical and experimental data sets from our and other research groups. In this way, we show data-based support of our hypothesis, which states that under conditions of food abundance incompetent brain-pull will lead to build-ups in the supply chain culminating in obesity and type 2 diabetes. In the same way, we demonstrate support of the related hypothesis, which states that under conditions of food deprivation a competent brain-pull mechanism is indispensable for the continuance of the brain s high energy level. In conclusion, we took the viewpoint of integrative physiology and provided evidence for the necessity of brain-pull mechanisms for the benefit of health. Along these lines, our work supports recent molecular findings from the field of neuroenergetics and continues the work on the "Selfish Brain" theory dealing with the maintenance of the cerebral and peripheral energy homeostasis.

Making sense of AMPA receptor trafficking by modeling molecular mechanisms of synaptic plasticity.

Synaptic plasticity involves a complex molecular machinery with various protein interactions but it is not yet clear how its components give rise to the different aspects of synaptic plasticity. Here we ask whether it is possible to mathematically model synaptic plasticity by making use of known substances only. We present a model of a multistable biochemical reaction system and use it to simulate the plasticity of synaptic transmission in long-term potentiation (LTP) or long-term depression (LTD) after repeated excitation of the synapse. According to our model, we can distinguish between two phases: first, a "viscosity" phase after the first excitation, the effects of which like the activation of NMDA receptors and CaMKII fade out in the absence of further excitations. Second, a "plasticity" phase actuated by an identical subsequent excitation that follows after a short time interval and causes the temporarily altered concentrations of AMPA subunits in the postsynaptic membrane to be stabilized. We show that positive feedback is the crucial element in the core chemical reaction, i.e. the activation of the short-tail AMPA subunit by NEM-sensitive factor, which allows generating multiple stable equilibria. Three stable equilibria are related to LTP, LTD and a third unfixed state called ACTIVE. Our mathematical approach shows that modeling synaptic multistability is possible by making use of known substances like NMDA and AMPA receptors, NEM-sensitive factor, glutamate, CaMKII and brain-derived neurotrophic factor. Furthermore, we could show that the heteromeric combination of short- and long-tail AMPA receptor subunits fulfills the function of a memory tag.

Deductive functional assignment of elements in appetite regulation.

This paper presents a simple mathematical model for energy transport from the body into the brain and for appetite regulation. Particular properties in appetite regulation are deduced from the general observation of cyclic food intake. These particular properties are the importance of a push component, however small it may be, from the body into the brain, the dependence of the appetite activation on the energy supply level in the brain and a necessary condition for the sensitivity of this dependence. The dominant pull component in the energy transport is accompanied by a smaller push component managing this information transport. The properties listed above correspond to physiological observations. For instance, sensitivity is found in the postnatal development of projections between neuropeptide Y (NPY) neurons and pro-opiomelanocortin (POMC) neurons, which release, respectively, the appetite-amplifying and -reducing neuropeptides NPY and alpha-melanocyte-stimulating hormone at their nerve terminals. The development of these projections determines the change from the permanent feeding of the foetus into the cyclic ingestive behaviour in later life. The correspondence verifies the mathematically-deduced properties, justifies the simple model and supports the technique of the deductive functional assignment.